Problem: $f(x)=\dfrac{1}{1+6x}$ Find a power series for $f$. Choose 1 answer: Choose 1 answer: (Choice A) A $1+6x+36x^2+\ldots +6^n x ^n+\ldots$ (Choice B) B $1+6x-36x^2+\ldots +(-6)^n x ^n+\ldots$ (Choice C) C $1-6x-36x^2+\ldots -6^n x ^{n}+\ldots$ (Choice D) D $1-6x+36x^2+\ldots +(-6)^n x ^{n}+\ldots$
Answer: This is a geometric series with first term $a\text{ }=\text{ }1$ and common ratio $r\text{ }=\text{ }-{6{x}}\,$. Therefore, the series is as follows. $1-6x+36x^2+\ldots +(-6)^n x ^{n}+\ldots $